**Date & Time:** Wednesday, 13th March, 2013 : 4.00 -5.00 p.m.

**Venue:** Ramanujan Hall

**Title:** From linear to piece wise linear modelling and back.
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**Speaker:** Prof. Andreas Griewank, Department of Mathematics, Humboldt University

**Abstract:** It is shown how functions that are defined by evaluation programs involving
: the absolute value function $\rabs()$ (besides smooth elementals), can be
: approximated locally by piecewise-linear models in the style of algorithmic,
: or automatic, differentiation (AD). The model can be generated by a minor
: modification of standard AD tools and it is Lipschitz continuous with respect
: to the base point at which it is developed. The discrepancy between the
: original function, which is {\em piecewise differentiable} and the piecewise
: linear model is of second order in the distance to the base point.
: Consequently, successive piecewise linearization yields bundle type methods
: for unconstrained minimization and Newton type equation solvers. As a third
: fundamental numerical task we consider the integration of ordinary
: differential equations, for which we examine generalizations of the midpoint
: and the trapezoidal rule for the case of Lipschitz continuous right hand
: sides. As a by-product of piecewise linearization, we show how to compute at
: any base point some generalized Jacobians of the original function, namely
: those that are {\em conically active} as defined by Khan and Barton.
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